Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
779 conjugacy classes of subgroups; only the 171 normal subgroups are shown.
This group has 779 conjugacy classes of subgroups — too many to draw. Only its 171 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(6,9)(7,11)(13,14)(15,17)(16,18)(19,20)(22,24)(23,27);(6,9)(10,12)(14,15)(18,19)(21,22,25,24)(23,28,27,26);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,15,17,14)(16,18,20,19)(21,23,25,27)(22,26,24,28);(5,7)(6,10)(8,11)(9,12)(13,16,17,20)(14,18,15,19)(21,24,25,22)(23,26,27,28)Polycyclic presentation:
2^8|a^2=fg,b^2=egh,c^2=g,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de52 conjugacy classes — view the full character table.
| maximal subgroup of | #103400, #103416, #103432, … |
| maximal normal subgroup of | #103400, #103416, #103432, … |
| maximal quotient of | #155755, #155845, #155869, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 9995);